Squares and Square Roots
Squares and square roots both concepts are opposite in nature to each other. Squares are the numbers, generated after multiplying a value by itself. Whereas square root of a number is value which on getting multiplied by itself gives the original value. Hence, both are vice-versa methods. For example, the square of 2 is 4 and the square root of 4 is 2.
If n is a number then its square is represented by n raised to the power 2, i.e., n^{2} and its square root is expressed as ‘√n’, where ‘√’ is called radical. The value under the root symbol is said to be radicand.
The square numbers are widely explained in terms of area of a square shape. The shape of a square is such that it has all its sides equal. Therefore, area of square is equal to (side x side) or side^{2}. Hence, if the side length of the square is 3cm then its area is 3^{2}= 9 sq.cm.
Properties of Square Numbers
The square numbers are the values which are produced when we multiply a number by itself. Some of the properties are:
- Square of 1 is equal to 1
- Square of positive numbers are positive in nature
- Square of negative numbers is also positive in nature. For example, (-3)^{2} = 9
- Square of zero is zero
- Square of root of a number is equal to the value under the root. For example, (√3)^{2} = 3
- The unit place of square of any even number will have an even number only.
- If a number has 1 or 9 in the unit’s place, then its square ends in 1.
- If a number has 4 or 6 in the unit’s place, then its square ends in 6.
Also, read:
Square Numbers 1 to 50
1^{2} = 1 |
11^{2} = 121 |
21^{2} = 441 |
31^{2} = 961 |
41^{2} = 1681 |
2^{2} = 4 |
12^{2} = 144 |
22^{2} = 484 |
32^{2} = 1024 |
42^{2} = 1764 |
3^{2} = 9 |
13^{2} = 169 |
23^{2} = 529 |
33^{2} = 1089 |
43^{2} = 1849 |
4^{2} = 16 |
14^{2} = 196 |
24^{2} = 576 |
34^{2} = 1156 |
44^{2} = 1936 |
5^{2} = 25 |
15^{2} = 225 |
25^{2} = 625 |
35^{2} = 1225 |
45^{2} = 2025 |
6^{2} = 36 |
16^{2} = 256 |
26^{2} = 676 |
36^{2} = 1296 |
46^{2} = 2116 |
7^{2} = 49 |
17^{2} = 289 |
27^{2} = 729 |
37^{2} = 1369 |
47^{2} = 2209 |
8^{2} = 64 |
18^{2} = 324 |
28^{2} = 784 |
38^{2} = 1444 |
48^{2} = 2304 |
9^{2} = 81 |
19^{2} = 361 |
29^{2} = 841 |
39^{2} = 1521 |
49^{2} = 2401 |
10^{2} = 100 |
20^{2} = 400 |
30^{2} = 900 |
40^{2} = 1600 |
50^{2} = 2500 |
Squares of Negative Numbers
The squares of negative numbers give a positive value, because if we multiply two negative numbers then it will result in a positive number.
Remember that: (-) x (-) = (+)
Therefore, square of (-n), (-n)^{2} = (-n) x (-n) = n^{2}
Where n is a number.
Examples:
- (-5)^{2} = (-5) x (-5) = 25
- (-7)^{2} = (-7) x (-7) = 49
Numbers between Squares
Suppose there are two square numbers n^{2} and (n+1)^{2}, then total numbers between these two squares are given by 2n.
Let’s say 3^{2} and 4^{2} are two squares.
3^{2} = 9 and 4^{2} = 16
We need to find the numbers present between 9 and 16.
Here, n = 3
Therefore, total numbers between 9 and 16 = 2n = 2 x 3 = 16
Is that correct? Let us check.
9, 10, 11, 12, 13, 14, 15, 16.
As we can see, the numbers between 9 and 19 are 6. Therefore, the formula given above is applicable to all the squares.
Numbers Between n^{2} and (n+1)^{2} = 2n, where n is any natural number |
Square Roots of Number
As we have already discussed, the square root of any number is the value which when multiplied by itself gives the original number. It is denoted by the symbol, ‘√’. If the square root of n is a, then a multiplied by a is equal to n. It can be expressed as:
√n = a then a x a = n
This is the formula for square root.
Square Roots of Perfect Squares
The perfect squares are the one whose square root gives a whole number. For example, 4 is a perfect square because when we take the square root of 4, it is equal to 2, which is a whole number. Let us see some of the perfect squares and their square roots.
Perfect Squares |
Square Root (√) |
0 |
0 |
1 |
1 |
4 |
2 |
9 |
3 |
16 |
4 |
25 |
5 |
Square Root of Imperfect Squares
Finding the square root of perfect squares is easy but to find the root of imperfect squares is difficult. The root of the perfect square can be estimated using the prime factorisation method.
The square root of imperfect squares is usually fractions. For example, 2 is an imperfect square because 2 cannot be prime factorised and its square root gives a fractional value.
Examples are:
- √2 = 1.4142
- √3 = 1.7321
- √8 = 2.8284
Square Roots 1 to 50
Number |
Square Root |
Number |
Square Root |
Number |
Square root |
√1 |
1 |
√18 |
4.2426 |
√35 |
5.9161 |
√2 |
1.4142 |
√19 |
4.3589 |
√36 |
6 |
√3 |
1.7321 |
√20 |
4.4721 |
√37 |
6.0828 |
√4 |
2 |
√21 |
4.5826 |
√38 |
6.1644 |
√5 |
2.2361 |
√22 |
4.6904 |
√39 |
6.2450 |
√6 |
2.4495 |
√23 |
4.7958 |
√40 |
6.3246 |
√7 |
2.6458 |
√24 |
4.8990 |
√41 |
6.4031 |
√8 |
2.8284 |
√25 |
5 |
√42 |
6.4807 |
√9 |
3 |
√26 |
5.0990 |
√43 |
6.5574 |
√10 |
3.1623 |
√27 |
5.1962 |
√44 |
6.6332 |
√11 |
3.3166 |
√28 |
5.2915 |
√45 |
6.7082 |
√12 |
3.4641 |
√29 |
5.3852 |
√46 |
6.7823 |
√13 |
3.6056 |
√30 |
5.4772 |
√47 |
6.8557 |
√14 |
3.7417 |
√31 |
5.5678 |
√48 |
6.9282 |
√15 |
3.8730 |
√32 |
5.6569 |
√49 |
7 |
√16 |
4 |
√33 |
5.7446 |
√50 |
7.0711 |
√17 |
4.1231 |
√34 |
5.8310 |
Frequently Asked Questions on Squares and square roots
What are squares and square roots?
Square numbers are the numbers which are produced when a value is multiplied by itself. Say if n is a number and is multiplied by itself, then the square of n is given by n^{2}. For example, the square of 10 is 10^{2} = 10 x 10 = 100.
The square root of a number is a value which on multiplied by itself gives the original number. It is represented by the symbol ‘√’. For example, the square root of 25 is √25 = 5.
What are perfect squares? Give an example
Perfect squares are the numbers, the square root of which gives a whole number. For example, 9 is a perfect square, because its root is a whole number, i.e.√9 = 3.
What is an imperfect square with examples?
An imperfect square is a number, the square root of which gives a fraction. The value generated by taking the square root of the imperfect square could be non-terminating as well.
For example, 3 is an imperfect square, because its root is equal to,
√3 = 1.73205080757, which is a fraction.
How to find the square root?
To find the square root of a number we can use the prime factorisation method. For example, the square root of 900 is:
√900 = √(2 x 2 x 3 x 3 x 5 x 5 )
Taking out the numbers in pairs, we get;
√900 = 2 x 3 x 5 = 30
900 was a perfect square, but to find the root of an imperfect square, we have to use long division method.
What is the difference between a square and square roots?
The square root of a number gives the root of a number which was squared. This is the primary difference between them.